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Full text of "Treatise On Analysis Vol-Ii"

21    PRODUCT OF MEASURES       219

(c) Show that there exists a measurable subset A of X with measure >0, such that for
each XQ e A there exists a function g e JS?(X, ju) (depending on x0) for which the
series with general term (g \fn)fn(x0) does not converge. (Take A to be the set of points
x e X for which the sequence (/(*)) does not converge to 0, and use Problem 7(a)
of Section 13.11. Then argue by contradiction, using (a) and (b).)

5.    Let S be a closed subset of R. Given a sequence (cn)n^0 of real numbers, show that
there exists a positive measure ju on R with support contained in S and such that

I x" dp(x) = cn for all n S> 0 if and only if, for every polynomial P(X) =  &X* such
J                                                                                                       * = o

that P(x) ;> 0 for all *sS, we have  f*c*^0. (As in Problem 2 of Section 13.3,

fc = 0

show that there exists a positive measure /JL with support S which extends to a linear
form u defined on #"R(R) and on the space of polynomials on R, and such that

w(P) = Z kCk for each polynomial P(X) =  & X*. Then prove that each power xn

* = 0                                                                                 fc=0

is /x-integrable and that J xn d^x) = cn. For this purpose, remark that for each in-
teger n ^ 0 and each  > 0 there exists a number R > 0 such that |/n,R(*)| <^ ex2n+2,
where /, R is the function equal to 0 for \x\ < R and to x* for \x\ ;> R.)

Particular cases: (1) S = R ("Hamburger's moment problem"): the condition is

n

that the quadratic forms    c/+*6* should be positive for ail n (observe that every

J.fcsaO

polynomial P(#) which is ^0 on R is the sum of two squares (P^x))2 + (P2Ct))2).
(2)   S = [0, 4- oo [ 0' Stieltjes* moment problem "): the condition is that the quad-

n                                          n

ratic forms  ]  Cj+kjk and ^ cj+k+i^j^k should be positive for all n> 0 (remark

J, *c = 0                                    J, *=0                                                                                                      ~~

that every polynomial P(x) which is i> 0 on [0, + oo [ can be written in the form
(Pi(x))2 4- (P2W)2 + *((P3(*))2 + (P^C*))2).

6.    Let v be a positive measure on X and (/) a sequence of functions belonging to

, v) such that the sequence of measures (fn-v) converges to f-v (where
, v)) for the topology ST^. Show that f(x) ^ lim sup/n(x) almost everywhere

with respect to v. (Use (13.8.3).)

7.    Let ju,,  v be two  bounded complex measures  on X. Show that the following
conditions  are  equivalent:   (1)   ^ and v are disjoint;  (2)   ||/uv|| = \\p\\ + ||v||;

||v||).

21. PRODUCT OF MEASURES

(13.21.1) Let X, Y be two locally compact spaces, 1 a measure on X and \i a
measure on Y. Then there exists a unique measure v on the product space
X x Y such that, for each pair of functions fe Jf C(X) andg G Jfc(Y), we have

(13.21.11) 0. (Reduce to the case where g is bounded and hence